Seek medical care. Cubital tunnel is compression of the ulnar nerve behind the inside of the elbow. Any activity that keeps the elbow bent can compress the nerve. The symptoms start with a funny feeling in the arm then intermittent numbness of the little and part of the ring finger starts. This can wake one at night. If ignored one can develope permanent numbness and hand weakness. See a hand surgeon.
Surgery. you can either try to rest the elbow and wrap towel around it when sleeping to prevent flexion but eventually you will need surgery. I do it endoscopically. Please see my youtube video on my technique: http://www.youtube.com/watch?v=X5A2g0JZrpo.